{
  "id": "1501.04079",
  "version": "v1",
  "published": "2015-01-16T19:09:06.000Z",
  "updated": "2015-01-16T19:09:06.000Z",
  "title": "Topology and convexity in the space of actions modulo weak equivalence",
  "authors": [
    "Peter Burton"
  ],
  "comment": "34 pages",
  "categories": [
    "math.DS"
  ],
  "abstract": "We analyse the structure of the quotient $\\mathrm{A}_\\sim(\\Gamma,X,\\mu)$ of the space of measure-preserving actions of a countable discrete group by the relation of weak equivalence. This space carries a natural operation of convex combination. We show that the convex structure of $\\mathrm{A}_\\sim(\\Gamma,X,\\mu)$ is compatible with the topology, and as a consequence deduce that $\\mathrm{A}_\\sim(\\Gamma,X,\\mu)$ is path connected. Using ideas of Tucker-Drob we are able to give a complete description of the topological and convex structure of $\\mathrm{A}_\\sim(\\Gamma,X,\\mu)$ for amenable $\\Gamma$ by identifying it with the simplex of invariant random subgroups. In particular we conclude that $\\mathrm{A}_\\sim(\\Gamma,X,\\mu)$ can be represented as a compact convex subset of a Banach space if and only if $\\Gamma$ is amenable. We consider the space $\\mathrm{A}_{\\sim_s}(\\Gamma,X,\\mu)$ of stable weak equivalence classes and show that is always a compact convex subset of a Banach space. For a free group $\\mathbb{F}_N$, we show that if one restricts to the compact convex set $\\mathrm{FR}_{\\sim_s}(\\mathbb{F}_N,X,\\mu) \\subseteq \\mathrm{A}_{\\sim_s}(\\mathbb{F}_N,X,\\mu)$ of the stable weak equivalence classes of free actions, the extreme points are dense in $\\mathrm{FR}_{\\sim_s}(\\mathbb{F}_N,X,\\mu)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-01-16T19:09:06.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37A35"
    ],
    "keywords": [
      "actions modulo weak equivalence",
      "stable weak equivalence classes",
      "compact convex subset",
      "convex structure",
      "banach space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 34,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150104079B"
    }
  }
}